


deceptive cadence

by luxaucupe



Series: terra firma [8]
Category: Wolf 359 (Radio)
Genre: Gen, and miranda but piano, concept:, this is basically just, what if hera but math
Language: English
Status: Completed
Published: 2019-08-29
Updated: 2019-08-29
Packaged: 2020-09-29 12:22:47
Rating: Teen And Up Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 1,718
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/20435963
Author URL: https://archiveofourown.org/users/luxaucupe/pseuds/luxaucupe
Summary: An excerpt fromFinite Differences,the second book penned by up-and-coming mathematician Hera. Her debut,Shallow Learning: How to Forgive the Math You’re Made Of,made history as the first ever book written entirely by an AI to reach number one on the New York Times Best Seller list.





	deceptive cadence

**Author's Note:**

> look, i think i've established myself enough in the w359 fic feed that i can just get weird with it. can i just get weird with it? i'm gonna get weird with it.  
aren't you tired of writing plots? don't you just wanna go apshit?  
that being said, here's four full fucking pages of me doing _math._
> 
> today's song recs (hell, as long as we're getting weird on here): porno disaster by the octopus project and my dog's eyes by zammuto

> An excerpt from _ Finite Differences _ , the second book penned by up-and-coming mathematician Hera. Her debut, _ Shallow Learning: How to Forgive the Math You’re Made Of, _ made history as the first ever book written entirely by an AI to reach number one on the _ New York Times _ Best Seller list.

* * *

_ Chapter 4 — Deceptive Cadence _

Let’s start at the beginning: I don’t know a damn thing about music. 

Actually, this is a bit of a lie. I technically know everything there is to know about music. It’s all right here, categorized neatly in my databases: the full Wikipedia article on the history of the lute, fourteen different music-related textbooks they supposedly use at Juilliard, the complete works of everyone from Bach to Billie Holiday to Radiohead. It means absolutely _ nothing _ to me. It’s like telling a little kid with an eidetic memory to read through your post-doc article on the role of endogenous viral elements in the development of the human genome. They’ve memorized the words, sure, but it might as well be gibberish. They haven’t _ learned _anything. And I’m no different.

What little I do actually, _ properly _ comprehend about music has been picked up slowly, bit-by-bit, by social osmosis. Some I’ve absorbed in larger chunks, mostly from the _ Star Wars _-laden ramblings of old friends (miss you, Doug). But it still feels distant to me at times. Untranslatably foreign.

My dear friend and roommate, Miranda, is a bit like running across the Rosetta Stone, in this sense.

Without going into too much detail, Miranda is a prodigy in many a field, mathematics and the general sciences among them. More recently, she has distanced herself from these talents, and I don’t blame her in the slightest. To her, they felt more like a legacy thrust upon her than an actual calling. 

Maybe once she loved them. The only thing that matters now is that, without feeling the need to provide an explanation, she doesn’t anymore, which means she has all the more love in her heart to feed into other passions.

Perhaps I should have mentioned earlier: Miranda is an exceptionally skilled pianist. 

Her skill is not a simple matter of my opinion. Having never actually learned to play an instrument or compose music, I have no frame of reference on the difficulty of these tasks beyond word of mouth. I still struggle to understand what makes a musician or a song inherently _ good _ . And yet, I can confidently say that she is among the better pianists of this century. I hear her play and something about it, something I can’t quite put my finger on, just feels _ thoughtful. _

She also routinely brings grown folk to tears with her compositions.

She’s _ that _ good.

And, as a pianist with a foundation in hard science, she has found a way to transcribe the more esoteric matters of her music into a language I speak fluently: _ math. _

Upon my somewhat joking request to explain the mere concept of an octave, she hand-wrote me a rather verbose yet surprisingly graceful paper detailing various mathematical aspects of music theory. This chapter is, in essence, my very long-winded, roundabout way of sharing said essay with you. Edited quite substantially for readability, mind you, but the credit (and dearest thanks) for the following belongs first and foremost to Miranda nonetheless.

This time, we’ll start somewhere in the middle. Musicians and mathematicians alike, I beg of you to bear with me while I undoubtedly get a bit redundant here. It’s not without very good reason.

A musical octave is a set of tones divided into eight notes, named A through G in Western music notation, such that any given note has half the vibrational frequency of the next note of the same name, thus sharing a number of auditory traits. Because of this octave equivalency, an octave can be treated cyclically, with the note after G returning back to A, and the cycle repeating. The note C, for example, behaves effectively the same whether it is played at 261.6 Hz or 523.2 Hz, though one is, of course, much lower than the other. 

In this manner, the notes of an octave behave a bit like the numbers of a clock. You count up the numbers on a clock, and once you hit twelve, they wrap around and start right back at one. With notes, you begin at A, ‘count’ up, and once you hit G, the next tone up is another A.

_ This _ is called _ modular arithmetic. _

This circle of frequencies can be cleanly cut into twelfths such that any one note and another are divided by interval spaces, named semitones, of ratio 2(1/12). _ But wait, _ I hear you cry! _ Twelve semitones and eight notes! _Fear not, dears. For those of you whose musical knowledge extends only to the trauma of being forced to learn recorder in grade school, this is why some notes come with those pesky sharps and flats.

All that mess may look like it has a lot of bullshit words. And, well, it does. But they’re also _ precise _words, and without precision, it gets harder to look at it all with genuine wonder.

For example, in these details (and with the addition of some background knowledge on the tuning of instruments), one might stumble upon the fact that it is actually impossible to tune a piano. Miranda has rambled endlessly on the subject, to my great appreciation, though I will leave those finer musings to those who are curious enough to look it up themselves.

Here’s the thing: no matter how _ neat _ one might find this all, it does a very poor job of proving that the _ idea _ of pretty music is a thing of math. And so we continue, delving just a bit deeper.

Two notes, when played together, may be either harmonious — each ringing pleasantly with the other, creating a sweet sound — or dissonant. Dissonant notes are still frequently played together in contemporary music, but dissonance has historically been considered unpleasant, and the sound produced by dissonance is often used in music to create a sense of tension to build up to a harmonious resolution. This harmony or dissonance is primarily decided by the number of semitone steps between the two notes, regardless of which beginning note is selected.

This definition that I’ve given here, to a diehard music theorist, has so many holes that Miranda has taken to calling it _ swiss cheese. _ But it’s short and sweet (as far as fancy, sweeping definitions go), so do me a solid and pretend I’ve written a proper one in its place. You wouldn’t like the look of the real definition.

I will conveniently hop over any attempt to fully define a chord, here, as well. I can only hope you all at least _ vaguely _ know what a chord is. Miranda’s original essay spend a full page on this bad boy. She’s nothing if not thorough, but I’ll spare you the labor. Let’s just call them groups of three notes that are all harmonious with each other, and go from there.

Imagine a circle.

Now, imagine all the little notes, from A to G and back again, with all their little sharps and flats. There are twelve, so we might as well stick them around that aforementioned circle like the numbers on a clock. If you humans have gone so far as to forget what an analog clock looks like, too bad. We’re sticking with the retro simile here.

In our very, _ very _ simplified definition of harmony, we can agree on the following: two notes are harmonious if they are either 3, 4, or 5 steps away from each other on the clock. ( _ Why those distances? _ I don’t know, and I honestly don’t care to find out. Miranda assures me it’s also purely mathematical. I’ll trust her word on this one.)

Back to our graph — err, _ clock. _ Let’s draw a line between every pair of harmonious notes. Now we have a neat little spiderweb-looking circle. Very fun. Super useless. Unless you know what you’re looking at.

For all the mathematicians and bored CS majors out there, this is a circulant graph by the name of _ C(12; 3, 4, 5). _ For everyone else, you can name it whatever you’d like. I’ve heard you’re all _ big _ fans of naming inanimate objects.

If we want to make music, we need chords. To get chords, we start with three notes that sound nice together, right? Here’s the good news: these chords show up as something special on our clock: triangles.

I don’t mean that as some bullshit special math or music term you need to google. I mean _ literal _ triangles. If each of three notes harmonizes with the other two, then they all harmonize together. Three points, three lines. A triangle. We have 28 to choose from.

Pick a triangle. Any triangle. Play those three notes together.

Congratulations. You’ve made _ music. _

_ Every single one _ of the 24 standard harmonious chords appears as a triangle on this graph.

But, oh, dear. Twenty-four is definitely less than twenty-eight. We have four extras. Math says these four triangles sound pretty together. Ears (and, y’know, auditory sensors) say they sound like shit. _ Math has failed us! Burn it all down! _

Not quite. Math has one more trick up its sleeve.

One good look at our note-clock tells us these triangles are all symmetrical — their notes are evenly spaced around the circle, and thus their sides are all the same length. In fact, they’re the _ only _ symmetrical ones. 

Conveniently enough, music has its own special definition for symmetrical. It’s the exact same as our math-y one: a group of notes that are evenly spaced around an octave. Symmetry in music comes with a notable side effect — any symmetrical triangle lacks the landmarks made by differently-lengthed sides, and thus doesn’t have any reliable method by which to define a “beginning” or “end”. The chord does not “belong” to any one note.

Hence, ears and math both agree that these four triangles sound _ bad. _

Pick a better triangle. One of the asymmetrical ones. Now, you’ve made music, _ and _ you know why it worked.

I’m not a musician. It’s not my trade.

But Miranda is, and she shared a little bit of that with me.

Thank you, Miranda, for the music.

**Author's Note:**

> yes, this is all real math, and mostly all real music theory. i got bored last year in math and did a bunch of useless research.  
comments are my meals. my delicious delicious fuel.  
and hey,  
thanks for reading.


End file.
